International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C, ch. 8.7, p. 725
Section 8.7.3.9. Uncertainties in derived functions^{a}732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^{b}Digital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^{c}Ecole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France |
The electrostatic moments are functions of the scale factor, the positional parameters x, y and z of the atoms, and their charge-density parameters κ and . The standard uncertainties in the derived moments are therefore dependent on the variances and covariances of these parameters.
If M_{p} represents the m × m variance–covariance matrix of the parameters , and T is an n × m matrix defined by for the lth moment with n independent elements, the variances and covariances of the elements of m^{l} are obtained from If a moment of an assembly of pseudo-atoms is evaluated, the elements of T include the effects of coordinate rotations required to transfer atomic moments into a common coordinate system.
A frequently occurring case of interest is the evaluation of the magnitude of a molecular dipole moment and its standard deviation. Defining where is the dipole-moment vector and G is the direct-space metric tensor of the appropriate coordinate system. If Y is the 1 × 3 matrix of the derivatives , where M_{μ} is defined by (8.7.3.96a). The standard uncertainty in μ may be obtained from σ(μ) = σ(y)/2μ. Significant contributions often result from uncertainties in the positional and charge-density parameters of the H atoms.